Question

Differentiate the function.

2. \(g\left( t \right) = ln\left( {3 + {t^2}} \right)\)

Short answer

The value of the derivative is \(g'\left( t \right) = \frac{{2t}}{{{t^2} + 3}}\).

Step-by-step solution

Use the derivative of logarithmic function

Rule 2: The derivative of \(\ln x\) is,

\(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\)

Evaluate the derivative of given function

Given that \(g\left( t \right) = \ln \left( {3 + {t^2}} \right)\)

Differentiate \(g\left( t \right)\)with respect to t,

\(\begin{aligned}{}\frac{d}{{dt}}g\left( t \right) &= \frac{d}{{dt}}\ln \left( {3 + {t^2}} \right)\\g'\left( t \right) &= \frac{{d\ln \left( {3 + {t^2}} \right)}}{{d\left( {3 + {t^2}} \right)}} \times \frac{{d\left( {3 + {t^2}} \right)}}{{dt}}\\g'\left( t \right) &= \frac{1}{{\left( {3 + {t^2}} \right)}} \times \left( {0 + 2t} \right)\\g'\left( t \right) &= \frac{{2t}}{{{t^2} + 3}}\end{aligned}\)

Thus, the value of the derivative is \(g\left( t \right) = \frac{{2t}}{{{t^2} + 3}}\).